A full approximation scheme multilevel method for nonlinear variational inequalities

TL;DR

This paper introduces the FASCD multilevel method, combining multigrid and constraint decomposition techniques, to efficiently solve various types of variational inequalities with nearly mesh-independent convergence.

Contribution

It extends the constraint decomposition approach by integrating it with the full approximation scheme multigrid method, achieving optimal convergence rates for complex variational inequalities.

Findings

FASCD V-cycles show nearly mesh-independent convergence.

Full multigrid cycles are optimal solvers for the tested problems.

Method applies to symmetric, nonsymmetric, and nonlinear operators.

Abstract

We present the full approximation scheme constraint decomposition (FASCD) multilevel method for solving variational inequalities (VIs). FASCD is a common extension of both the full approximation scheme (FAS) multigrid technique for nonlinear partial differential equations, due to A.~Brandt, and the constraint decomposition (CD) method introduced by X.-C.~Tai for VIs arising in optimization. We extend the CD idea by exploiting the telescoping nature of certain function space subset decompositions arising from multilevel mesh hierarchies. When a reduced-space (active set) Newton method is applied as a smoother, with work proportional to the number of unknowns on a given mesh level, FASCD V-cycles exhibit nearly mesh-independent convergence rates, and full multigrid cycles are optimal solvers. The example problems include differential operators which are symmetric linear, nonsymmetric linear, and nonlinear, in unilateral and bilateral VI problems.